Years ago, I used to teach informal reasoning. One of the things I came to realize was that my students and I were in much the same position when it came to names of fallacies: I'd get myself to memorize them during the term, but not long after, I'd forget most of the names, just as my students presumably did. Still, I think that in this case we can come up with a name that may even be helpful.
Start here: the conclusion is a complete non sequitur ; it doesn't even remotely follow from the premises. How do we get from "The poor in some countries are worse off than the poor in our country" to "The poor in our country should be immiserated until they are as wretched as the poor in those other countries"?
Notice that the premise is a bald statement of fact, while the conclusion tells use what we ought to do about the fact. By and large, an "ought" doesn't simply follow from an "is", and so we have a classic "is/ought" fallacy. However, pointing this out isn't really enough. After all, in some cases...

The answer will vary with the sort of premise. For example: we confirm the truth of a mathematical claim in a very different way than we confirm the truth of a claim about the weather. Some things can be confirmed by straightforward observation (there's a computer in front of me). Some can be confirmed by calculation (for example, that 479x368=176,272). Depending on our purposes and the degree of certainty we need, some can be confirmed simply by looking things up. (That's how I know that Ludwig Wittgenstein was born in 1889.) Some call for more extensive investigation, possibly including the methods and techniques of some scientific discipline. The list goes on. It even includes things like appeal to consensus, when the consensus is of people who have relevant expertise. I'm not a climate scientist. I believe that humans are contributing to climate change because the consensus among experts is that it's true. But the word "expert" matters there. The fast that a group of my friends happen to think that...

When we look at arguments, we have two broad questions in mind. One is whether the conclusion follows from the premises, whether or not the premises are true. The other is whether the premises are actually true. So with that in mind, let's turn to the argument.
It's often possible by restating premises and adding other premises that are assumed but not stated to make an argument valid even if it's not valid as stated. Your argument is more or less this, I think
If God exists, then necessarily God is perfectly good, knows all, and is all-powerful,
Suppose God exists.
Since God is all-powerful, God can give us free will.
Since God is perfectly good, God wants us to have free will.
God does anything God wants to do.
Therefore, we have free will.
Since God knows all, God knows what we are going to do before we do it.
If God knows what we're going to do before we do it, then we don't have free will.
Therefore, we don't have free will.
CONTRADICTION.
Therefore, God doesn't exist.
We could clean things...

I'm going to step through this carefully to make sure I follow.
We have two theories: A and B . Theory A has an assumption: A and theory B has an assumption B . And A and B are mutually exclusive—can't both be true.
Let's pause. To say that a theory has an assumption means that if the theory is true, then the assumption is true. It doesn't mean that if the assumption is true, then the theory is true. A silly example: the special theory of relativity assumes that objects can move in space. But from the assumption that objects can move in space, the special theory of relativity doesn't follow; you need a lot more than that. Otherwise, the "assumption" would be the real theory.
You ask if it's true that neither theory can prove anything about the other. If I understand the question aright, it's not true. For one thing, trivially, if we take A as a premise, then by your own description, it follows that B is false. That seems like a case of proving something about B ...

Fun question.
Let's say that a characteristic or property or whatnot is intrinsic if we can tell whether someone has it without needing information about other people/things. The fact that I have blue eyes is an intrinsic feature in that sense. My eye color doesn't depend on your eye color. But to know that I'm the shortest person in the room, you have to know things about the other people in the room as well as things about me (namely, our heights.) Being the shortest person in the room isn't an intrinsic property/quality/characteristic. Note that we're using "property", "characteristic", "quality" so as to include abstract things, and things that depend in possibly quite recondite ways on how an individual is related to other individuals, sets of individuals...
We don't tend to use the word feature so abstractly. Your features are the things we'd talk about to describe you yourself. Some of them, like height, may not be purely intrinsic, but to make things simple, we'll set those aside.
If we...

I don't have Baggily and Fosl's book handy but if your quote is accurate, there's clearly a mistake—almost certainly a typo or proof-reading error. The tautology that goes with a valid argument is the hypothetical whose antecedent is the conjunction of the premises and whose consequent is the conclusion. Thus, if
P, Q therefore R
is valid, then
(P & Q) → R
is a tautology, or better, a truth of logic. So if the text reads as you say, good catch! You found an error.
However, your question suggests that you're puzzled about how a valid argument could be stated as a tautology at all. So think about our example. Since we've assumed that the argument is valid, we've assumed that there's no row where the premises 'P' and 'Q' are true and the conclusion 'R' false. That means: in every row, either 'P & Q' is false or 'R' is true. (We've ruled out rows where 'P & Q' true and 'R' is false.) So the conditional '(P & Q) → R' is true in every row, and hence is a truth of logic.

I think you're getting at the difference between the principle of Bivalence (there are only two truth values—true and false) and the Law of Excluded Middle: 'P or not-P' is always true.
Suppose there are some sentences that are neither true nor false. That might be because they are vague, for example. It might not be true to say that Smith is bald, but it might not be false either; it might be indeterminate. So if S stands for "Smith is bald," then "Either S is true or S is false" would not be correct. Our assumption is that S isn't true, but also isn't false. However, if by "not- S " we mean " S isn't true," then " S or not- S " is true. That is, bivalence would fail, but excluded middle wouldn't.
But as you might imagine, there's a good deal of argument about the right thing to say here.

I'm not entirely sure I follow, but perhaps this will be of some use. Whether two propositions really have four possible combinations of truth values depends on the propositions. Non-philosophical examples make the point easier to follow. Suppose P is "Paula is Canadian" and Q is "Quincy is Australian." In this case, the two propositions are logically independent, and all four combinations P&Q, P&~Q, ~P&Q and ~P&~Q represent genuine possibilities. But not all propositions are independent in this way; it depends on their content. P and Q might be contradictories, that is, one might be the denial of the other. (If P means that Paula is Canadian and Q means that she is not Canadian, then we have this situation.) In that case, the only two possibilities are P&~Q and ~P&Q. Or P and Q might be contraries, meaning that they can't both be true though they could both be false. For example: if P is "Paula is over 6 feet tall" and Q is "Paula is under 5 feet tall," then we only have three...

I think you're right to at least this extent. If I say to someone "I won all the games of chess I played," the normal rules of conversation (in particular, the "pragmatics" of speech) make it reasonable for the other person to infer that I have actually played at least one game. Whether my statement literally implies this, however, is trickier. Think about statements of the form "All P are Q." Although it may take a bit of reflection to see it, this seems to be equivalent to saying that nothing is simultaneously a P and a non-Q. We can labor the point a bit further by turning to something closer to the lingo of logic: there does not exist an x such that x is a P and also a non-Q. For example: all dogs are mammals. That is, there does not exist a dog that is a non-mammal. Now go back go the games. If Mary says "All games I played are games I won," then by the little exercise we just went through, this becomes "There does not exist a game that I played and lost." But if Mary played no games at all,...

More or less every textbook I can think of has many, many translations of symbolic sentences into English. Many, though by no means all, of the translations are in the exercises, and often you need to work from answer to question, but any good text will include lots and lots of examples. What I mean by "work from answer to question", by the way, is this: the more common kind of symbolization problem goes from English into symbols. The question will give you the English sentence, and the answer—often at the end of the chapter—will give the symbolic version. But if you look at the answer and trace it back to the question, you have just what you want. The question might ask you to put "No man is his own brother" into symbols. The answer might look like this: ~∃x(Mx ∧ Bxx) But if you are given the answer and you know what question it answered, then you have your translation. Bear in mind that for this to work, you have to know what the letters stand for; that's often given in the question....